Theorem cotr3 15027

Description: Two ways of saying a relation is transitive. (Contributed by RP, 22-Mar-2020.)

Hypotheses

Ref	Expression
cotr3.a	⊢ 𝐴 = dom 𝑅
cotr3.b	⊢ 𝐵 = (𝐴 ∩ 𝐶)
cotr3.c	⊢ 𝐶 = ran 𝑅

Assertion

Ref	Expression
cotr3	⊢ ((𝑅 ∘ 𝑅) ⊆ 𝑅 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧))

Distinct variable groups:   𝑥,𝑦,𝑧,𝑅   𝑥,𝐴,𝑦,𝑧   𝑦,𝐵,𝑧   𝑧,𝐶

Allowed substitution hints:   𝐵(𝑥)   𝐶(𝑥,𝑦)

Proof of Theorem cotr3

Step	Hyp	Ref	Expression
1		cotr3.a	. . 3 ⊢ 𝐴 = dom 𝑅
2	1	eqimss2i 4070	. 2 ⊢ dom 𝑅 ⊆ 𝐴
3		cotr3.b	. . . 4 ⊢ 𝐵 = (𝐴 ∩ 𝐶)
4		cotr3.c	. . . . 5 ⊢ 𝐶 = ran 𝑅
5	1, 4	ineq12i 4239	. . . 4 ⊢ (𝐴 ∩ 𝐶) = (dom 𝑅 ∩ ran 𝑅)
6	3, 5	eqtri 2768	. . 3 ⊢ 𝐵 = (dom 𝑅 ∩ ran 𝑅)
7	6	eqimss2i 4070	. 2 ⊢ (dom 𝑅 ∩ ran 𝑅) ⊆ 𝐵
8	4	eqimss2i 4070	. 2 ⊢ ran 𝑅 ⊆ 𝐶
9	2, 7, 8	cotr2 15026	1 ⊢ ((𝑅 ∘ 𝑅) ⊆ 𝑅 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1537  ∀wral 3067   ∩ cin 3975   ⊆ wss 3976   class class class wbr 5166  dom cdm 5700  ran crn 5701   ∘ ccom 5704

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ral 3068  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-br 5167  df-opab 5229  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711

This theorem is referenced by: (None)